Density of linear combination

Question

Let $r_1, \ldots, r_n$ be a set of positive reals. Define \begin{equation*} S = \{a_1r_1+\cdots+a_nr_n : a_i\in \mathbb{N}\}. \end{equation*} Define $\pi(x)= |\{a\in S:a<x\}|$. Is there an asymptotic formula for $\pi(x)$ as $x \to \infty$ ? For $n=1$, $\pi(x) \approx x/a_1 $. But I'm pretty unsure how to generalize this.

Answer 1

I think you can recast your problem in such a way that $\pi(x)$ is given by a (Quasi) Ehrhart Polynomial.

Quoting from that wiki page: In mathematics, an integral polytope has an associated Ehrhart polynomial that encodes the relationship between the volume of a polytope and the number of integer points the polytope contains. The theory of Ehrhart polynomials can be seen as a higher-dimensional generalization of Pick's theorem in the Euclidean plane.